iPhone 4S, iPad 2, iPad 3rd generation, and earlier devices (iPhone 4, iPhone 3GS, iPad 1, and iPod touch 3rd and 4th generation) running iOS 5.1.1 can be jailbroken untethered using Absinthe or redsn0w — both tools provide the same jailbreak. redsn0w may work more reliably; if one tool gives you error messages and does not successfully jailbreak your device, try the other tool. Please backup your device with iTunes before jailbreaking, just in case anything goes wrong.

Download Absinthe 2.0.4 to your desktop and run it - see Chronic Dev Team for downloads for OS X, Windows, and Linux.

If Absinthe isn't working: try backing up your data, restoring your device with iTunes, and attempting the jailbreak again. On Windows, make sure to run Absinthe as administrator and in XP compatibility mode (as described below for redsn0w), make sure you're opening the right file, and temporarily disable any antivirus/firewall programs you have running. If things still aren't working, you might consider using a different computer to jailbreak your device.

OS X Mountain Lion: You have to run Absinthe from terminal on Mountain Lion. Right click the Absinthe icon and select “Show Package Contents”. Navigate to the Contents/MacOS folder and right click on Absinthe 10.6 and select Open With…Terminal.

Unzip redsn0w and run it. Do not run it from inside the zip; instead, extract it to the desktop. OS X users, skip to the next section. For Windows users: You will need to run it as Administrator, in XP Service Pack 3 (2 if on Vista) compatibility mode. Do this by right-clicking the redsn0w.exe file, Properties, Compatibility tab, select Windows XP (Service Pack X [choose 3 if there, otherwise 2]). Then check the box at the bottom labeled Run this program as an administrator. It should look something like this screenshot. Click OK. Also try temporarily disabling any antivirus or firewall programs, such as AVG or Avast, if redsn0w doesn't seem to work. Windows XP only:
Click redsn0w.exe on your desktop once to select it, and then hold down the Shift key and right-click redsn0w.exe. You'll see a contextual menu from which you can select Run as... and then choose an administrator account.

For non-A5 devices (iPhone 4, iPhone 3GS, iPad 1, and iPod touch 3rd/4th generation), redsn0w will give you the choice of using the Absinthe-style method (which does not require DFU mode but takes a long time if the device has a lot of data on it) or the traditional limera1n-based ramdisk method (which includes checkboxes for installing Cydia and/or the iPad baseband, and which "hacktivates" your non-A5 device if it is in an unactivated state). Both methods provide the same jailbreak.

Click the "Jailbreak" button in redsn0w. Your device needs to be connected to your computer at this point. At the next step, choose "Install Cydia" only, and follow the instructions.

After the jailbreak is complete, open up Cydia and wait for it to download the latest package data from its repositories - you'll know it's done after "Reloading Data" shows up for a few seconds and disappears. Then you can install anything you like!

iPhone 3GS notes: If you're trying to jailbreak an iPhone 3GS and redsn0w doesn't work, make sure you've clicked the correct choice when redsn0w asks if your device has an "old" or "newer (fixed)" bootrom. (In other words, try the process again and choose the other option.) If you're trying to jailbreak an iPhone 3GS and also make it unlockable using the iPad baseband, redsn0w may crash if you install both Cydia and the iPad baseband at the same time. If you're seeing this problem, you can work around it by going through the jailbreaking process once just installing Cydia, and then a second time just installing the iPad baseband.

Run sn0wbreeze. Browse to (or drag and drop) the stock IPSW file onto sn0wbreeze. You should get a picture of your device and it should say "Verified." Click next.

If you use a software unlock (ultrasn0w), and don't have the original carrier's sim for activation, choose Expert, select "Activate the phone (hacktivate)," leave everything else as-is, and hit the next arrow twice. Otherwise, choose "Simple."

Select "Build IPSW." This will take a while. You can play Pac-Man or just wait.

When the build is done, sn0wbreeze will walk you through the process of putting your device into PWNED DFU mode. This is different from normal DFU and is necessary when restoring custom firmware.

Open iTunes. Click OK to the prompt saying your device is in recovery mode (iTunes calls this DFU recovery mode). Hold the shift key and click the Restore button in iTunes. Navigate to your desktop and select the sn0wbreeze IPSW file. This will take several minutes. When it's finished, you are jailbroken.

Apple TV 2G can be jailbroken untethered on 4.4.3 and 4.4.4 and 4.0-4.3.
Apple TV 2G only has a tethered jailbreak on 4.4-4.4.2 and on 5.1/5.0.
Apple TV 3G cannot be jailbroken at all and can not be exploited with developer exploits- the attack surface is smaller as ATV has no support for other applications other then AppleTV.app (on a non-jailbroken state)

since 5.0/5.1 has a completely different GUI than 4.x, few main applications are not yet compatible with it (XBMC for 5.1 is in nightly builds currently; Overflow, Plex and Remote HD are currently not compatible. All others are compatible)
The first package you should install is NitoTV (install only after jailbreak and installing OpenSSH). Here is a tutorial: http://iphoneblogr.com/2010/10/how-to-install-nitotv-on-your-jailbroken-apple-tv-2g/ .

To stitch ATV firmwares with blobs you can right click the Choose Firmware button in SP And choose a firmware and blobs.